No abstract

The power structure over the Grothendieck (semi)ring of complex quasi-projective varieties constructed by the authors is used to express the generating series of classes of Hilbert schemes of zero-dimensional subschemes on a smooth quasi-projective variety of dimension d as an exponent of that for the complex affine space A d . Specializations of this relation give formulae for generating series of such invariants of the Hilbert schemes of points as the Euler characteristic and the Hodge-Deligne polynomial.

In this work we present several examples of surface singularities in ~3, for which the p-constant stratum S,, in the miniversal deformation is not smooth.To do that we study surface singularities (V,0)c(IE3,0) that can be resolved by a quadratic transformation (to simplify we call such singularities superisolated (S.I.S.)). We study in w 1 #-constant deformations p: N-~ T of a S.I.S. with a smooth base T, and using the Perron's theorem [-11] and results of Neuman we prove in Theorems 1 and 2 that for such a deformation N is equimultiple along a(T) (a is the section of p) and p has a strong simultaneous resolution, starting with the monoidal transformation with center a(T). We see also that if DcIP z is the projectivized tangent cone of (V,0), then p induces a deformation n: ~T of D, which is equisingular as a deformation of the projective plane curve D c IP 2.In the Sect. 2 we study how to compute in the base (B) of a miniversal deformation of (V,0), the stratum #-constant (resp. V*-constant) which we denote by S u (resp. Su,). In order to do that we consider in B the equimultiplicity stratum E and the deformation of D over E induced by the miniversal deformation. Next we consider the stratum 2;DEE of the points where the corresponding plane curve has the same equisingularity type in its singularities as D. The main result is:In this way we reduce the problem of the computation of the p-constant stratum to the computation of the equisingularity stratum of a deformation of a projective plane curve, with possibly many singularities. Now we have developed an effective method to compute the equations of the equisingularity stratum in a deformation of a plane curve singularity, so that the equations of S~ can be computed; the only problem is the big size of the equations that may appear. We note that the methods give also a constructive proof of the smoothness of the #-constant stratum in a versal deformation of a plane curve singularity.

The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function Z DL (h, T ) of a quasi-ordinary power series h of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent Z DL (h, T ) = P (T )/Q(T ) such that almost all the candidate poles given by Q(T ) are poles. Anyway, these candidate poles give eigenvalues of the monodromy action of the complex of nearby cycles on h −1 (0). In particular we prove in this case the monodromy conjecture made by Denef-Loeser for the local motivic zeta function and the local topological zeta function. As a consequence, if h is a quasi-ordinary polynomial defined over a number field we prove the Igusa monodromy conjecture for its local Igusa zeta function.

For a germ of a meromorphic function f = P Q , we offer notions of the monodromy operators at zero and at infinity. If the holomorphic functions P and Q are non-degenerated with respect to their Newton diagrams, we give an analogue of the formula of Varchenko for the zeta-functions of these monodromy operators.